HMMT 十一月 2015 · 冲刺赛 · 第 24 题

英文解析

24. [ 12 ] Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of thethree edges connecting to the vertex they are on with equal probability and travel to the other vertexon that edge. They all stop when any two ants reach the same vertex at the same time. What is the probability that all three ants are at the same vertex when they stop?
    Proposed by: Anna Ellison
    Answer:1
    At every second, each ant can travel to any of the three vertices they are not currently on. Given that,16
    at one second, the three ants are on different vertices, the probability of them all going to the same
    1 11
    vertex is and the probability of them all going to different vertices is , so the probability of the
    27 27
    11 1
    th n − 1
    three ants all meeting for the first time on the n step is ( ) × . Then the probability the three
    27 27
    ∑1
    ∞
    11 i 1 1
    ants all meet at the same time is ( ) × = = .27
    i =011
    27 27 16
    1 −
    27